Skip to main content
SpringerLink
Log in

Twist Character of the Fourth Order Resonant Periodic Solution

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.-N. Chow C.Z. Li D. Wang (1994) Normal Forms and Bifurcation of Planar Vector Fields Cambridge University Press Cambridge

    Google Scholar 

  2. J. Lei X. Li P. Yan M. Zhang (2003) ArticleTitleTwist character of the least amplitude periodic solution of the forced pendulem SIAM J. Math. Anal 35 844–867 Occurrence Handle10.1137/S003614100241037X

    Article  Google Scholar 

  3. J. Lei M. Zhang (2002) ArticleTitleTwist property of periodic motion of an atom near a charged wire Lett. Math. Phys 60 9–17 Occurrence Handle10.1023/A:1015797310039

    Article  Google Scholar 

  4. J. Mawhin (1998) Nonlinear complex-valued differential equations with periodic, Floquet or nonlinear boundary conditions. International Conference on Differential Equations (Lisboa, 1995) World Scientific River Edge, NJ 154–164

    Google Scholar 

  5. Moser J. (1962). On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II, 1–20.

  6. D. Núñez (2002) ArticleTitleThe method of lower and upper solutions and the stability of periodic oscillations Nonlinear Anal 51 1207–1222 Occurrence Handle10.1016/S0362-546X(01)00888-4

    Article  Google Scholar 

  7. D. Núñez R. Ortega (2000) ArticleTitleParabolic fixed points and stability criteria for nonlinear Hill’s equations Z. Angew. Math. Phys 51 890–911

    Google Scholar 

  8. D. Núñez P.J. Torres (2001) ArticleTitlePeriodic solutions of twist type of an earth satellite equation Discr. Cont. Dynam. Syst 7 303–306

    Google Scholar 

  9. D. Núñez P.J. Torres (2003) ArticleTitleStable odd solutions of some periodic equations modeling satellite motion J. Math. Anal. Appl 279 700–709 Occurrence Handle10.1016/S0022-247X(03)00057-X

    Article  Google Scholar 

  10. R. Ortega (1992) ArticleTitleThe twist coefficient of periodic solutions of a time-dependent Newton’s equation J. Dynam. Differential Equations 4 651–665 Occurrence Handle10.1007/BF01048263

    Article  Google Scholar 

  11. R. Ortega (1994) ArticleTitleThe stability of the equilibrium of a nonlinear Hill’s equation SIAM J. Math. Anal 25 1393–1401 Occurrence Handle10.1137/S003614109223920X

    Article  Google Scholar 

  12. R. Ortega (1996) ArticleTitlePeriodic solution of a Newtonian equation: Stability by the third approximation J. Differential Equations 128 491–518 Occurrence Handle10.1006/jdeq.1996.0103

    Article  Google Scholar 

  13. Ortega, R. (2003). The stability of the equilibrium: a search for the right approximation, Preprint: Available at http://www.ugr.es/ ecuadif/fuentenueva.htm

  14. C.L. Siegel J.K. Moser (1971) Lectures on Celestial Mechanics Springer-Verlag Berlin

    Google Scholar 

  15. P.J. Torres M. Zhang (2004) ArticleTitleTwist periodic solutions of repulsive singular equations Nonlinear Anal 56 591–599 Occurrence Handle10.1016/j.na.2003.10.005

    Article  Google Scholar 

  16. P. Yan M. Zhang (2003) ArticleTitleHigher order nonresonance for differential equations with singularities Math. Methods Appl. Sci 26 1067–1074 Occurrence Handle10.1002/mma.413

    Article  Google Scholar 

  17. M. Zhang (2001) ArticleTitleThe rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials J. London Math. Soc 64 IssueID2 125–143

    Google Scholar 

  18. M. Zhang (2003) ArticleTitleThe best bound on the rotations in the stability of periodic solutions of a Newtonian equation J. London Math. Soc 67 IssueID2 137–148 Occurrence Handle10.1112/S0024610702003939

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, China

    Jinzhi Lei

  2. Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain

    Pedro J. Torres

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Meirong Zhang

Authors
  1. Jinzhi Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro J. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meirong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinzhi Lei.

Additional information

Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lei, J., Torres, P.J. & Zhang, M. Twist Character of the Fourth Order Resonant Periodic Solution. J Dyn Diff Equat 17, 21–50 (2005). https://doi.org/10.1007/s10884-005-2937-4

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-005-2937-4

Keywords

Search

Navigation